The Refractive Index of Curved Spacetime: the Fate of Causality in QED

TL;DR

This paper calculates the full frequency-dependent refractive index in curved spacetime, showing causality is preserved at high frequencies despite violations of dispersion relations, raising questions about micro-causality in quantum field theory.

Contribution

It provides the first full frequency dependence of the refractive index in curved spacetime using world-line sigma models and Penrose limits, demonstrating causality preservation at high frequencies.

Findings

Wavefront velocity equals c, ensuring causality.

Kramers-Kronig relations are violated due to non-analyticity.

Micro-causality may be challenged in curved spacetime QFT.

Abstract

It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime. Assuming the validity of the Kramers-Kronig dispersion relation, this would imply a superluminal wavefront velocity and the violation of causality. Here, we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic. We find that the high-frequency limit of the phase velocity (i.e. the wavefront velocity) is always equal to c and causality is assured. However, the Kramers-Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane, whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points. This puts into question the issue of micro-causality, i.e. the vanishing of commutators of field operators at spacelike separated points, in local quantum field theory in curved spacetime.